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Simple(?) theoretical set theory questions
Posted:
Jun 13, 1996 10:20 AM


I'm involved in a bit of discussion over in alt.algebra.help and thought I'd try to clarify some things we've been discussing.
Q1: What is the commonly accepted definition of two sets having the same cardinality? Is it "the existence of a bijection between them" ?
Q2: Do you need the Axiom of Choice to prove that every infinite set has a countable subset? My feeling is no, since the following should work: Let A be the set. A is nonempty so choose a_1 in A (no axiom of choice needed, a_1 exists since A is nonempty). Now A\{a_1} is infinite (otherwise A would be finite) and hence a_2 exists in A\{a_1}. Repeat (ie.induction) to get a_n for all natural N, ie. a countable subset.
Q3: Do you need the Axiom of Choice to show the following: For any two sets A and B, either there exists an injection from A to B or an injection from B to A
Thanks in advance, Dan



